Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T13:40:07.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Some applications of ordinal dimensions to the theory of differentially closed fields

Published online by Cambridge University Press:  12 March 2014

Wai Yan Pong
Wai Yan Pong*
Affiliation:
University of Illinoisat Chicago, 322 Science and Engineering Office (SEO) M/C 249, 851 S. Morgan Street, Chicago, IL 60607, USA, E-mail: pong@math.uic.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the Lascar inequalities, we show that any finite rank δ-closed subset of a quasiprojective variety is definably isomorphic to an affine δ-closed set. Moreover, we show that if X is a finite rank subset of the projective space ℙn and a is a generic point of ℙn, then the projection from a is injective on X. Finally we prove that if RM = RC in DCF0, then RM = RU.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 347 - 356
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Hodges, W., Model theory, Encyclopedia of mathematics and its applications, vol. 42, Cambridge University Press, 1993.CrossRefGoogle Scholar
[2]Hrushovski, E. and Scanlon, T., Lascar rank Morley rank differ in differentially closed fields, to appear in Journal of Symbolic Logic.Google Scholar
[3]Hrushovski, E. and SokoloviĆ, Z., Minimal subsets of differentially closed fields, to appear in Transactions of the American Mathematics Society.Google Scholar
[4]Johnson, J., Differential dimension polynomials and a fundamental theorem on differential modules, American Journal of Mathematics, vol. 91 (1969), pp. 239–248.CrossRefGoogle Scholar
[5]Johnson, J., A notion of Krull dimension of differential rings, Commentarii Mathematici Helvetici, vol. 44 (1969), pp. 207–216.CrossRefGoogle Scholar
[6]Kolchin, E., Differential algebra and algebraic groups, Pure and Applied Mathematics, a series of monographs and textbooks, vol. 54, Academic Press, 1973.Google Scholar
[7]Lascar, D., Ranks and definability in superstable theories, Israel Journal of Mathematics, vol. 23 (1976), pp. 53–87.CrossRefGoogle Scholar
[8]Lascar, D., Stability in model theory, Pitman monographs and surveys in pure and applied mathematics, vol. 36, Longman, 1987.Google Scholar
[9]Marker, D., Model theory of differential fields, Lecture Notes in Logic 5, ch. II, pp. 38–113, Lecture Notes in Logic 5, Springer Verlag, 1996, pp. 38–113.CrossRefGoogle Scholar
[10]McGrail, T., Model-theoretic results on ordinary and partial differential fields, Ph.D. thesis, Wesleyan University, 1997.Google Scholar
[11]Pillay, A., Differential Algebraic Groups and the Number of Countable Differentially Closed Fields, Lecture Notes in Logic 5, ch. III, pp. 114–134, Lecture Notes in Logic 5, Springer Verlag, 1996, pp. 114–134.Google Scholar
[12]Pong, W.Y., Ordinal dimensions and differential completeness, Ph.D. thesis, University of Illinois at Chicago, 1999.Google Scholar
[13]Saffe, J., Categoricity and ranks, this Journal, vol. 49 (1984), no. 4, pp. 1379–1392.Google Scholar